Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 30
Find all real solutions of the equation. $$ x^{2}+5 x-6=0 $$
Short Answer
Expert verified
The real solutions are \( x = 1 \) and \( x = -6 \).
Step by step solution
01
Identify the Quadratic Equation
The given equation is a quadratic equation in the standard form. A quadratic equation in standard form looks like \( ax^2 + bx + c = 0 \). Here, the equation is \( x^2 + 5x - 6 = 0 \) with \( a = 1 \), \( b = 5 \), and \( c = -6 \).
02
Apply the Quadratic Formula
The quadratic formula is used to find the solutions of a quadratic equation and is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. We will now apply this formula to our equation.
03
Calculate the Discriminant
First, calculate the discriminant \( b^2 - 4ac \) to check the nature of the roots. Substitute \( a = 1 \), \( b = 5 \), and \( c = -6 \) into the discriminant formula: \[ b^2 - 4ac = 5^2 - 4 \times 1 \times (-6) = 25 + 24 = 49 \]. Since 49 is positive, there are two distinct real roots.
04
Find the Roots
Substitute the values into the quadratic formula: \[ x = \frac{-5 \pm \sqrt{49}}{2 \times 1} = \frac{-5 \pm 7}{2} \].
05
Simplify to Find the Solutions
Calculate the two potential solutions: 1. \( x = \frac{-5 + 7}{2} = \frac{2}{2} = 1 \) 2. \( x = \frac{-5 - 7}{2} = \frac{-12}{2} = -6 \). Thus, the solutions are \( x = 1 \) and \( x = -6 \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It is applicable when the equation is in the standard form \( ax^2 + bx + c = 0 \). The formula is:
The beauty of the quadratic formula is its ability to provide solutions directly, whether they be real or complex numbers. To use this formula:
- \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The beauty of the quadratic formula is its ability to provide solutions directly, whether they be real or complex numbers. To use this formula:
- Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
- Calculate the discriminant \( b^2 - 4ac \).
- Substitute these values into the formula to find the solutions \( x \).
Discriminant
The discriminant is a key component of the quadratic formula that determines the nature of the roots of the equation. It is given by the expression \( b^2 - 4ac \). The value of the discriminant can tell us:
In our example equation \( x^2 + 5x - 6 = 0 \), we calculated the discriminant to be 49, which is positive, indicating two distinct real roots.
- If it is positive, the equation has two distinct real roots.
- If it is zero, there is exactly one real root (a double root).
- If it is negative, the equation has no real roots but two complex roots.
In our example equation \( x^2 + 5x - 6 = 0 \), we calculated the discriminant to be 49, which is positive, indicating two distinct real roots.
Real Roots
Real roots are solutions to a quadratic equation that are real numbers, meaning they are not imaginary. An equation can have:
For our example, because the discriminant was 49, the equation \( x^2 + 5x - 6 = 0 \) had two real roots, found using the quadratic formula: \( x = 1 \) and \( x = -6 \). These values satisfy the equation when substituted back in.
- Two distinct real roots (when the discriminant is positive).
- One real root (when the discriminant is zero, referred to as a repeated or double root).
For our example, because the discriminant was 49, the equation \( x^2 + 5x - 6 = 0 \) had two real roots, found using the quadratic formula: \( x = 1 \) and \( x = -6 \). These values satisfy the equation when substituted back in.
Standard Form of Quadratic Equation
The standard form of a quadratic equation is a specific way of writing the equation as \( ax^2 + bx + c = 0 \). This form is necessary for easily applying the quadratic formula and analyzing the equation.
To identify whether an equation is in standard form, ensure that:
In the exercise, the equation \( x^2 + 5x - 6 = 0 \) matches this standard form with \( a = 1 \), \( b = 5 \), and \( c = -6 \). This standardization makes solving with the quadratic formula straightforward.
To identify whether an equation is in standard form, ensure that:
- All terms are on one side of the equation, with zero on the other side.
- The equation is arranged in descending powers of \( x \).
- Coefficients \( a \), \( b \), and \( c \) are correctly aligned with their respective terms.
In the exercise, the equation \( x^2 + 5x - 6 = 0 \) matches this standard form with \( a = 1 \), \( b = 5 \), and \( c = -6 \). This standardization makes solving with the quadratic formula straightforward.
